Finite Group Databasebeta

← #20281929

Subgroups — #20281929

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810559C211yesmaximal, characteristic, complemented
2#16810557C211yesmaximal, characteristic, complemented
3#16810553C211yesmaximal, characteristic, complemented
4#16810545C211yesmaximal, characteristic, complemented
5#16810545C211yesmaximal, characteristic, complemented
6#16810523C211yesmaximal, characteristic
7#16810509C211yesmaximal, characteristic, complemented
8#16810504C211yesmaximal, characteristic, complemented
9#16810500C211yesmaximal, characteristic
10#16810499C211yesmaximal, characteristic
11#16810498C211yesmaximal, characteristic, complemented
12#16810495C211yesmaximal, characteristic, complemented
13#16810482C211yesmaximal, characteristic, complemented
14#16810480C211yesmaximal, characteristic, complemented
15C53×#399C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C2×C4⋊C4×C53C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C23×D53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C2×C22⋊D53—41no—
60C2×C22⋊D53—41no—
61C2×C22⋊D53Q814yes—
62C22×Dic53—41no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—41no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—22no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊Dic53—22no—
82C22⋊Dic53—41no—
83#11719921—41no—
84#11719921—41no—
85#11719921—41no—
86#11719921—41no—
87#11719921—41no—
88C4⋊Dic53—41no—
89C4⋊Dic53—41no—
90C4⋊Dic53—41no—
91#11719919—22no—
92#11719919—41no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96C4×Dic53—41no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53C2×Q812yes—
104C23×C53C2411yescharacteristic
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53C2×Q812yes—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53#4112yes—
130C22⋊D53—44no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—44no—
151C2×Dic53C2×Q812yes—
152C2×Dic53#4112yes—
153C4×D53—44no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×Q811yescharacteristic
162C22×C53C2×#4111yescharacteristic
163C22×C53C2×#4111yescharacteristic
164C22×C53H(5,𝔽2)11yescharacteristic
165C22×C53H(5,𝔽2)11yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C4×C53—22no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181Dic53—22no—
182Dic53—22no—
183Dic53—41no—
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188#2748—531nomaximal, Sylow
189C2×C53—42no—
190C2×C53#54011yescharacteristic
191C2×C53#54011yescharacteristic
192C2×C53#54311yescharacteristic
193C2×C53#55111yescharacteristic
194C2×C53#55411yescharacteristic
195C2×C53#55411yescharacteristic
196C2×C53#56011yescharacteristic
197D53—42no—
198#399—531no—
199#399D5311yescharacteristic, complemented
200#398—531no—
201#398—531no—
202#396—531no—
203#396—531no—
204#390—531no—
205#387—531no—
206#387—531no—
207#386—531no—
208#386—531no—
209#385—531no—
210#385—531no—
211#380—531no—
212#380—531no—
213C53#274811yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C23×C4—531no—
216C23×C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4C2×D5311yescharacteristic
226C2×C4⋊C4C2×D5311yescharacteristic
227C2×C4⋊C4C2×D5311yescharacteristic
228C2×C22⋊C4—531no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4—531no—
234C2×C22⋊C4—531no—
235C2×C22⋊C4C2×D5311yescharacteristic
236C2×C22⋊C4C2×D5311yescharacteristic
237C2×C22⋊C4C2×D5311yescharacteristic
238C2×C42—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95C2×D5311yescharacteristic
249C24—531no—
250C24C22×D5311yescharacteristic
251C2×D4—534no—
252C2×D4—2121no—
253C2×D4—2121no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—2121no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4—2121no—
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—1062no—
277C4⋊C4—2121no—
278C4⋊C4—2121no—
279C4⋊C4—2121no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—1062no—
288C22⋊C4—2121no—
289C22⋊C4—2121no—
290C22⋊C4—2121no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C42—2121no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—532no—
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23Q8×D5312yes—
309C23C23×D5311yescharacteristic
310D4—2124no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—532no—
329C2×C4—532no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—2124no—
350C2×C4—2124no—
351C2×C4#1171994712yes—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—1062no—
360C22—1062no—
361C22—1062no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22C2×Q8×D5311yescharacteristic
367C22#41×D5311yescharacteristic
368C22C2×#1171994711yescharacteristic
369C22#1313326211yescharacteristic
370C22#1313326211yescharacteristic
371C22#1313326311yescharacteristic
372C22#1313326311yescharacteristic
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—1062no—
380C4—1062no—
381C4—2121no—
382C4—2121no—
383C4—2121no—
384C4—2121no—
385C4—2121no—
386C2—42no—
387C2—2122no—
388C2#1681122511yescharacteristic
389C2#1681122711yescharacteristic
390C2#1681124811yescharacteristic
391C2#1681129611yescharacteristic
392C2#1681132711yescharacteristic
393C2#1681133111yescharacteristic
394C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2748
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #16810559C2semidirect
2. #16810557C2semidirect
3. #16810553C2semidirect
4. #16810545C2semidirect
5. #16810545C2semidirect
7. #16810509C2semidirect
8. #16810504C2semidirect
11. #16810498C2semidirect
12. #16810495C2semidirect
13. #16810482C2semidirect
14. #16810480C2semidirect
15. C53×#399C2semidirect
199. #399D53semidirect
213. C53#2748semidirect